# Effective annual interest rate

Effective annual interest rate |
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**The effective annual interest rate** is the annual rate of interest equivalent to a given interest rate. It depends on the nominal interest rate and periods in which interest is capitalized, i.e. on the frequency of capitalization.

The effective annual interest rate is the percentage by which the value of capital is *increased* in *one year*.

## Formula for effective annual interest rate

We calculate the effective annual interest rate from the formula

Where:

- - effective annual interest rate
- - nominal annual interest rate
- - number of capitalization per year (e.g. for semi-annual capitalization m = 2, quarterly m = 4, monthly m = 12)

As it results from the formula, the **effective** annual interest rate **depends** primarily on:

- nominal annual interest rate
- the number of capitalization in a year - present both in the exponent of power and in the denominator of a fraction

## Conclusions resulting from pattern analysis

At the fixed nominal rate, taking into account the dependence of the effective rate on the annual interest rate, we can formulate the following conclusions:

- the effective rate is equal to the nominal rate only with annual capitalization
- the effective rate is higher than the nominal rate if the capitalization period is shorter than one year
- the effective rate is the higher the more interest is capitalized
- the effective rate is the highest with continuous capitalization.

## Examples of Effective annual interest rate

**Compound Interest**: Compound interest is the interest accrued on a principal amount of money over time, where the interest is added to the principal, and the interest rate is applied to the new, larger, principal. The effective annual interest rate is the rate of interest that is equivalent to the compounded interest rate over the course of one year.**Mortgage/Loan Interest**: When taking out a loan or mortgage, the effective annual interest rate takes into account the frequency of when the interest is capitalized (monthly, quarterly, semi-annually, etc). The effective annual interest rate is the rate of interest that is equivalent to the loan/mortgage interest rate over the course of one year.**Credit Card Interest**: Credit cards typically have a variable interest rate that is applied to any outstanding balance that is carried over from month to month. The effective annual interest rate is the rate of interest that is equivalent to the credit card interest rate over the course of one year.

## Advantages of Effective annual interest rate

The effective annual interest rate offers several advantages:

- It allows for comparison of different interest rates with different capitalization periods. This allows for a more accurate comparison of different financial options, such as mortgages, loans and investments.
- It makes it easier to calculate the total cost of borrowing, as it takes into account the frequency of compounding. This makes it easier for borrowers to compare different loan products and make the best choice for their needs.
- It helps to estimate the future value of money invested by taking into account the effect of compounding. This allows investors to make better decisions regarding the timing and size of investments.

## Limitations of Effective annual interest rate

The effective annual interest rate has several limitations when it comes to estimating the actual return on an investment. These limitations include:

- The effective annual interest rate only considers the nominal interest rate and does not factor in any other costs associated with the investment, such as transaction fees or annual maintenance fees.
- It does not take into account the effects of inflation, which can erode the value of the investment over time.
- It is based on the assumption that interest is compounded annually, which may not be the case in reality.
- It does not account for any changes in the nominal interest rate, which can significantly impact the return on an investment.
- It does not consider the time value of money, which means that the return on an investment is not adjusted for the length of time that the investment is held.

An effective annual interest rate is a measure of the true cost of borrowing money over time. Other approaches related to effective annual interest rate include the following:

**Compound interest rate**: This is the rate at which interest is compounded over a given period of time, usually one year. This rate takes into account the frequency and amount of interest payments and can be used to calculate the effective annual interest rate.**Nominal interest rate**: This is the stated rate of interest, without taking into account the frequency of compounding.**Annual percentage rate (APR)**: This is the annual rate of interest charged on a loan or credit card, including any fees and other costs.**Real interest rate**: This is the annual rate of interest after taking into account inflation.

In summary, an effective annual interest rate is an important measure of the true cost of borrowing money over time. It is related to other approaches such as the compound interest rate, nominal interest rate, APR, and real interest rate.

## References

- Clark, M. W. (1994).
*A note on the calculation of the effective annual rate for cash discounts*. Journal of Accounting Education, 12(1), 77-79. - Thomas, A. L. (1968).
*Estimating the Effective Interest Rate*. The Accounting Review, 43(3), 589-591. - Pournara, C. (2015).
*Effective interest rates: making sense or cents?*. Learning and Teaching Mathematics, 2015(18), 46-50.